01/08/2019
“Textbook” sample size caclulation for a normal endpoint:
\[ \begin{aligned} \min_{n \in \mathbf{N}} ~ & n \\ \text{subject to } ~ & g(n, \mu, {\color{red}{\sigma}}) \geq {\color{blue}{1 - \beta^*}} \end{aligned} \] \(g(n, \mu, {\color{red}{\sigma}})\) - power of the trial.
\({\color{red}{\sigma}}\) - an unknown nuisance parameter.
\({\color{blue}{1 - \beta^*}}\) - a power threshold.
Minimial Clinically Imortant Difference: \(\mu = 0.3\)
Power threshold: \(1 - \beta^* = 0.8\)
Universe A: \(\hat{\sigma} = 1 \rightarrow n = 175\)
Universe B: \(\hat{\sigma} = 1.3 \rightarrow n = 296\)
Same effect to be detected, same power, but different sample size.
More generaly, as nuisnace parameter varies, so does the amount we are willing to invest in a study to get 80% power to detect \(\mu = 0.3\).
No flexibility to samba - already declared \(\mu = 0.3\).
\(\rightarrow\) Sample size re-estimation is incoherent.
We should:
This would:
Choose \(n\) to maximise value, denoted \(v(n, \sigma)\), a weighted sum of power and sample size:
\[ \max_{n} v(n, \sigma) = g(n, \sigma) - \lambda n \]
Implicit assumptions about value:
Two arm parralel group trial comparing group means of normally distributed outcome.
Difference to detect: 0.3
Best guess of standard deviation: 1
Trade-off parameter \(\lambda\): 0.0022
(\(\rightarrow n = 175\) under both frameworks)
\(\rightarrow\) A value-based approach will lead to a coherent framework for sample size re-estimation, with less variability in \(n\) but more variability in power.
But, we can go further - in some cases, we don’t need to do re-estimation at all.
\[ n = n_i(\sigma_t) [1 + (m-1)\rho] \] \(n_i(\sigma_t)\) = sample size for an individually randomised trial with no clustering and the same total variance, \(\sigma_t^2\).
\(\rho =\) Intacluster correlation coeffieint (ICC) - proportion of the total variance due to variability between clusters.
\(m =\) number of participants per cluster
$k = $ number of clusters
Number of clusters: 15 Number of participants: 470
Number of clusters: 18 Number of participants: 500
Summary
Further work